Pre-Calc and recursive formula?

Question

I'm in Honors Pre-Calculus and we're going over some things we learned in Algebra 2. I'm having a lot of trouble gripping it (just like I did in Algebra 2). I'm going over recursive formula and "U sub 1" is how it's described in the book. I have an example from a previous quiz that I bombed.. could someone explain how recursive formula works and go step by step through this problem with me?

Problem #6:
Dante deposited $10,000 into an account with an annual interest rate of 3.75%, compounded quarterly. He plans to deposit $750 each quarter just after the interest is compounded.

a. Write a recursive formula for Dante.

b. What will his balance be after 5 years?

I have no idea how to write a recursive formula or figure it out.. I'm completely lost and swamped with over-due homework because I couldn't do it.

Any help will be greatly appreciated..
Thank you in advance. :]

Unfortunately.. none of that made ANY sense to me..

Answer 1

The basic idea behind a recursive formula is that you have to repeat a number of similar steps a few times in order to get the answer. The solution at each step, depends on the solution from the preceding step.

Applied to your specific problem: Dante's balance after 5 years depends on what his balance is after 4 years.

But his balance after 4 years depends on what his balance was after 3 years.

And so on.

You could use the variable "U5" to stand for his balance after 5 years, and "U4" to stand for his balance after 4 years. Every quarter he adds $750 and every quarter he gets 0.9375% interest (that's 3.75% divided by 4):

Balance at the end of 4 years = U4
Balance at the end of 4.25 years = (U4+750)(.9375%)
Balance at the end of 4.5 years = ((U4+750)(.9375%) + 750)(.9375%)
Balance at the end of 4.75 years = (((U4+750)(.9375%) + 750)(.9375%) + 750)(.9375%)
Balance at the end of 5 years = U5 = ((((U4+750)(.9375%) + 750)(.9375%) + 750)(.9375%) + 750)(.9375%)

We used U5 and U4 in the example, but you can make the same statement about any year "n" and the year before, "n-1":

If "U_sub_n" is the amount after "n" years, and "U_sub_n_1" is the amount after "n-1" years, then:

U_sub_n = ((((U_sub_n_1 + 750)(.9375%) + 750)(.9375%) + 750)(.9375%) + 750)(.9375%)

That's basically the recursive formula.

To figure out the answer for U5, first figure out U4.
To figure out U4, first figure out U3.
To figure out U3, first figure out U2.
To figure out U2, first figure out U1.
To figure out U1, first figure out U0. (Hint: U0 = $10000.)

Answer 2

Just take what is happening bit by bit...

A recursive formula is just about describing the current event f(k) in a chain of events in terms of the previous event f(k-1).

Think of each quarter as being a new iteration

f(k) = the sum in the bank just after Dante has made his kth deposit

We are told that he earns an annual interest rate of 3.75% compounded quarterly.
To find out how much he gets each quarter we need to find the value X such that (X)^4 = 1.0375
Taking logs of both sides and factoring we get log(X) =log(1.0375)/4
=> X = 1.00925 (or 0.925% per quarter).

To work out f(k) - however much money Dante had after the previous quarter has an additional 0.9246% added to it, then Dante adds another $750

So f(k) = f(k-1)*1.009246 + 750

We are also told that his very first deposit is $10,000. This is f(0)

So Part a) is f(k) = 1.009246f.(k-1) + 750 with f(0)=10,000

E.g straight after his first $750 deposit (3 months after opening account) he will have
1.009246*10,000 + 750 = $10842.46 = f(1)


For part b) you need to find f(20) = amount in account after 20 quarters (5 years).

You can just work out f(1), f(2), f(3) etc - which would be easy to do in something like Excel: have a column for k (k=0 to 20) and a column for f(k) and calculate the f(k) value from the previous value in the column above - using the formula we have already found. Remembering to just put in a value of 10,000 against f(0).

Doing the above I get f(20) = 28414.61073 = $28414.61

HOWEVER - if we round every time we recalculate to the nearest cent (as a bank would do)
i.e. f(k) = ROUND[f(k-1)*1.009246 + 750, 2 dp] - again very easy to do with excel
We actually get f(20) = $28414.59 - only a 2 cent difference, but more realistic to what would happen in a bank!

I know it's quite a lot of steps to go through - but think about what you need as you go along
1) calculate quarterly interest
2) define f(k) in terms of f(k-1)
3) set a starting value for f(0)
4) work forwards again to get the value of k you need


Sometimes with recursives you get lucky - and you can automatically define f(k) in terms of just k and f(0) to give a nice easy answer without needing to do all the iterations. Unfortunately in this case there is no simple formula and you end up with (Q=quarterly interest rate 1.09246)

f(k) = f(0)*Q^20 + 750*(Q^19 + Q^18 + Q^17 + ... + Q^2 + Q + 1)

Unfortunately this does not simplify further....


If you are saying this makes no sense - have you even covered recursive formulae in your class yet. I was assuming a slight level of knowledge (since you had homework on the subject).

Answer 3

A recursive formula is a formula where you write the "new value" in terms of the "old value". You keep plugging in the values "recursively". An initial value is usually supplied too, so that you can get the values started.

In this case, the new balance is the old balance plus 3.75% of it, plus another $750. So you could say that if his balance after q quarters (after the new deposit) is B_q ("B sub q"), then

B_q = 750 + ( B_[q-1] * 1.0375)
B_1 = 10,000

So...
B_5 = 750 + (B_4 * 1.0375), where
B_4 = 750 + (B_3 * 1.0375), etc.

Answer 4

sadly..i don't remember either..and i'm in honors pre-calculus too..but we're doing f of g of x stuff..sorry...if i remembered i would help..